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Scale 2999: "Diminishing Nonamode"

Scale 2999: Diminishing Nonamode, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names

Western Modern
Diminishing Nonamode
Western Mixed
Chromatic and Permuted Diatonic Dorian Mixed
Zeitler
Zyrygic

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

9 (enneatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,2,4,5,7,8,9,11}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

9-11

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

none

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

yes
enantiomorph: 3515

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

6 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

3 (tricohemitonic)

Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

2

Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

8

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 1775

Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 2, 1, 2, 1, 1, 2, 1]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<6, 6, 7, 7, 7, 3>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p7m7n7s6d6t3

Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2}
<2> = {2,3}
<3> = {3,4,5}
<4> = {5,6}
<5> = {6,7}
<6> = {7,8,9}
<7> = {9,10}
<8> = {10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

1.111

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

yes

Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

2.799

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

6.106

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.

none

Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Proper

Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(0, 51, 138)

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsC{0,4,7}342.39
C♯{1,5,8}442.22
E{4,8,11}442.28
F{5,9,0}342.44
G{7,11,2}342.67
A{9,1,4}342.44
Minor Triadsc♯m{1,4,8}442.17
dm{2,5,9}342.56
em{4,7,11}342.5
fm{5,8,0}442.17
am{9,0,4}342.39
Augmented TriadsC+{0,4,8}542
C♯+{1,5,9}442.33
Diminished Triadsc♯°{1,4,7}242.67
{2,5,8}242.67
{5,8,11}242.56
g♯°{8,11,2}242.72
{11,2,5}242.72
Parsimonious Voice Leading Between Common Triads of Scale 2999. Created by Ian Ring ©2019 C C C+ C+ C->C+ c#° c#° C->c#° em em C->em c#m c#m C+->c#m E E C+->E fm fm C+->fm am am C+->am c#°->c#m C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->d° C#->fm dm dm C#+->dm F F C#+->F C#+->A d°->dm dm->b° em->E Parsimonious Voice Leading Between Common Triads of Scale 2999. Created by Ian Ring ©2019 G em->G E->f° g#° g#° E->g#° f°->fm fm->F F->am G->g#° G->b° am->A

view full size

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2999 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3547
Scale 3547: Sadygic, Ian Ring Music TheorySadygic
3rd mode:
Scale 3821
Scale 3821: Epyrygic, Ian Ring Music TheoryEpyrygic
4th mode:
Scale 1979
Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic
5th mode:
Scale 3037
Scale 3037: Nine Tone Scale, Ian Ring Music TheoryNine Tone Scale
6th mode:
Scale 1783
Scale 1783: Youlan Scale, Ian Ring Music TheoryYoulan Scale
7th mode:
Scale 2939
Scale 2939: Goptygic, Ian Ring Music TheoryGoptygic
8th mode:
Scale 3517
Scale 3517: Epocrygic, Ian Ring Music TheoryEpocrygic
9th mode:
Scale 1903
Scale 1903: Rocrygic, Ian Ring Music TheoryRocrygic

Prime

The prime form of this scale is Scale 1775

Scale 1775Scale 1775: Lyrygic, Ian Ring Music TheoryLyrygic

Complement

The enneatonic modal family [2999, 3547, 3821, 1979, 3037, 1783, 2939, 3517, 1903] (Forte: 9-11) is the complement of the tritonic modal family [137, 289, 529] (Forte: 3-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2999 is 3515

Scale 3515Scale 3515: Moorish Phrygian, Ian Ring Music TheoryMoorish Phrygian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2999 is chiral, and its enantiomorph is scale 3515

Scale 3515Scale 3515: Moorish Phrygian, Ian Ring Music TheoryMoorish Phrygian

Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 2999       T0I <11,0> 3515
T1 <1,1> 1903      T1I <11,1> 2935
T2 <1,2> 3806      T2I <11,2> 1775
T3 <1,3> 3517      T3I <11,3> 3550
T4 <1,4> 2939      T4I <11,4> 3005
T5 <1,5> 1783      T5I <11,5> 1915
T6 <1,6> 3566      T6I <11,6> 3830
T7 <1,7> 3037      T7I <11,7> 3565
T8 <1,8> 1979      T8I <11,8> 3035
T9 <1,9> 3958      T9I <11,9> 1975
T10 <1,10> 3821      T10I <11,10> 3950
T11 <1,11> 3547      T11I <11,11> 3805
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 4019      T0MI <7,0> 2495
T1M <5,1> 3943      T1MI <7,1> 895
T2M <5,2> 3791      T2MI <7,2> 1790
T3M <5,3> 3487      T3MI <7,3> 3580
T4M <5,4> 2879      T4MI <7,4> 3065
T5M <5,5> 1663      T5MI <7,5> 2035
T6M <5,6> 3326      T6MI <7,6> 4070
T7M <5,7> 2557      T7MI <7,7> 4045
T8M <5,8> 1019      T8MI <7,8> 3995
T9M <5,9> 2038      T9MI <7,9> 3895
T10M <5,10> 4076      T10MI <7,10> 3695
T11M <5,11> 4057      T11MI <7,11> 3295

The transformations that map this set to itself are: T0

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 2997Scale 2997: Major Bebop, Ian Ring Music TheoryMajor Bebop
Scale 2995Scale 2995: Raga Saurashtra, Ian Ring Music TheoryRaga Saurashtra
Scale 3003Scale 3003: Genus Chromaticum, Ian Ring Music TheoryGenus Chromaticum
Scale 3007Scale 3007: Zyryllian, Ian Ring Music TheoryZyryllian
Scale 2983Scale 2983: Zythyllic, Ian Ring Music TheoryZythyllic
Scale 2991Scale 2991: Zanygic, Ian Ring Music TheoryZanygic
Scale 2967Scale 2967: Madyllic, Ian Ring Music TheoryMadyllic
Scale 3031Scale 3031: Epithygic, Ian Ring Music TheoryEpithygic
Scale 3063Scale 3063: Solyllian, Ian Ring Music TheorySolyllian
Scale 2871Scale 2871: Stanyllic, Ian Ring Music TheoryStanyllic
Scale 2935Scale 2935: Modygic, Ian Ring Music TheoryModygic
Scale 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic
Scale 2487Scale 2487: Dothyllic, Ian Ring Music TheoryDothyllic
Scale 3511Scale 3511: Epolygic, Ian Ring Music TheoryEpolygic
Scale 4023Scale 4023: Styptyllian, Ian Ring Music TheoryStyptyllian
Scale 951Scale 951: Thogyllic, Ian Ring Music TheoryThogyllic
Scale 1975Scale 1975: Ionocrygic, Ian Ring Music TheoryIonocrygic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy.